Heat Conduction Problems of Thermosensitive Solids under Complex Heat Exchange

139
the surface temperature of the body with constant characteristics. The latter temperature is
to be found from the problem:

div
g
rad Po ( , , ,Fo)
Fo
H
H
T
TqXYZ



, (38)


Bi ( 0
H
Hc
s
T
TT
n




H
H
T
T






, (41)

()
0
H
s
T
n




, (42)

Fo 0
()
H
p
TT


()
s
T

of the thermosensitive body in the condition (27) is
equal to the corresponding temperature of the body with constant characteristics, then the
boundary value problem for the Kirchhoff’s variable

should be solved with the condition
(33). Then the solution of this problem presents the difference of the temperature in the
same-shape body with constant characteristics and the initial temperature:

H
p
TT



. (44)
As it was mentioned above, the substitution of
()T

for
p
T


in the case of linear
dependence of the heat conductivity coefficient on the temperature is equivalent to keeping
only two terms in the series, into which the square root in expression for the temperature

to linearize the problem completely by linearizing the nonlinear condition on
Kirchhoff’s variable

obtained from condition of convective heat exchange due to
replacement of nonlinear expression
()T

by (1 )
p
T



 with unknown parameter

;
-
to solve the obtained linear boundary value problem for variable

by means of an
appropriate classical method;
-
to satisfy with given accuracy the nonlinear condition for variable

by using the
parameter

;
-
to determine the temperature using the obtained Kirchhoff’s variable.

characteristics in the coupled elements. By making use of the Kirhoff’s integral
transformation for each element by assuming the thermal conductivity to be constants, the
problem can be partially linearized. The nonlinearities remain due to the thermal contact
conditions on the interfaces and the conditions of complex heat exchange on the surfaces. To
obtain an analytical solution to the coupled problem for the Kirchhoff’s variable, it is
necessary to linearize this problem. The possible ways of such a linearization and, thus,
determination of the general solution to the heat conduction problems in piecewise-
homogeneous bodies are considered below in this section.

Heat Conduction Problems of Thermosensitive Solids under Complex Heat Exchange

141
Let us adopt the method of linearizing parameters to solution of the steady-state heat
conduction problems for coupled bodies of simple shape, for instance,
n -layer
thermosensitive cylindrical pipe. The pipe is of inner and outer radii
0
rr

and
n
rr ,
respectively, with constant temperatures
b
t and
H
t on the inner and outer surfaces. The
layers of different temperature-dependent heat conduction coefficients are in the ideal
thermal contact. The cylindrical coordinate system
,,rz

tttt



 , (46)

() ( 1)
1
11
,() ()
ii
ii
ii ti t i
dt dt
tt t t
dr dr





, =, 1, 1
i
rri n

 , (47)
where
()
()
i

functions,
0
t is the reference temperature. In the dimensionless form, the problem (45)–(47)
appears as

()
1
() 0, 1,
i
i
ti
dT
d
Tin
dd

 





, (48)

1
1
,
n
bn H
TTT T



. (50)
Consider the heat conduction coefficients in the form of linear dependence on the
temperature
()
()
0
() (1 )
i
i
ti ii
t
tkT

, where
i
k are constants. By introducing the Kirchhoff’s
variable

()
0
()
i
T
i
it
TdT






 



 , (53)





11 1
() ( 1)
1
00
12 1/ 12 1/ ,
at , 1, 1
ii i i i i
i
ii
ii
tt
kk k k
in
dd
dd



()
0
()
T
n
t
TdT





.
The initially nonlinear heat conduction problem is partially linearized due to application of
the Kirchhoff’s variables. However, the conditions for temperature, that reflects the
temperature equalities of the neighbouring layers, remain nonlinear (the first group of
conditions (54)). By integrating the set of equations (52) with boundary conditions (53) and
contact conditions (54), the set of transcendent equation can be obtained for determination
of constant of integration. This set can be solved numerically. The efficiency of numerical
methods depends on the appropriate initial approximation. Unfortunately, it is very
complicated to determine the definition domain for the solution of this set of equations and
thus to present a constructive algorithm for determination of the initial approximation.
The possible way around this problem is to decompose the square root in the first
conditions (54) into series by holding only two terms. Then, instead of mentioned
conditions, the following approximated conditions are obtained:

1
at , 1, 1
ii i
in

i
d
d
dd


 






, (58)

1
1
,
n
bn n


 

 


 , (59)

1

t


, 1, 1in

 .

Heat Conduction Problems of Thermosensitive Solids under Complex Heat Exchange

143
It can be shown (Podsdrihach et al., 1984) that the boundary value problem (58)–(60) is
equivalent to the problem

1
() 0
d
d
dd

 
 






, (61)

1

1
12
1
1
ln 1 1
ln ( )
()
n
jj
jj
j
CSC


  









 







 
  





 

















 

















  









;
1
1
1
11
ln
i


н
1
1
1
()
1
0
() ( 1) ()
1
0
00
11
1
(1 ) (1 ) ln ln
n
jj
i
n
in b n j
t
njj
j
t
tt
A


    


0
(1) ()
1
00
11
1
(1 ) ln
1
i
jj
i
ibi
j
t
jj
i
j
tt
BA


 










, for instance, is equal to zero, the
following set of 1n

equations can be obtained





11 1
12 1/ 12 1/ , 1, 1
ii
ii i i i i
kkk kin
 

 


   (67)
for determination of the rest 1n

linearizing parameters. The solution should be found in a
neighborhood of zero. From the set (67), we determine the values of linearization
parameters and thus the Kirchhoff’s variables. Then the temperature in layers is

1
(1 2 1)
ii ii





,
н
н2
2
2
1
1
(1 )
ln
(1 )ln ln
b
K

 











, (69)



 

 











 


н
н
1
2
22112
(1 )
1
12 ln 1

i

 are constants, then the
temperature in each layer is determined by formula

н12
ln , ln
b
TNK TTN T




, (71)
where



н
(2) (1)
12
(1)lnln,
btt
NTT K K




    .
Let the first layer of thickness

 [( )]WmK

,
(2)
t


64.5(1 0.49 T)
[( )]WmK . Then
1
0.37k

 ,
(1)
0
47.5
t

 ,
2
0.49k

 ,
(2)
0
64.5
t

 , 1.36K




0
(1) (2)
tc tc



(1) (2)
00tt



T
Ct


T
Ct


T
Ct


T
Ct


1 1 700 1 700 1 700 1 700

values in the fifth and sixth columns describe the case when the heat conduction coefficients
have the mean value in the temperature region 0 700 C i.e.
700
(1) (1)
0
1
( ) 38.7
700
tc t
tdt




[( )]WmK ,
(2)
1
700
tc


700
(2)
0
() 48.7
t
tdt




conditions on Kirchhoff’s variables may be fulfilled using the method of linearizing
parameters.
The method of linearizing parameters can be successfully used for solution of the transient
heat conduction problems.
5. Determination of the temperature fields by means of the step-by-step
linearization method
To illustrate the step-by-step linearization method, consider the solution of the centro-
symmetrical transient heat conduction problem. Let us consider the thermosensitive hollow
sphere of inner radius
1
r and outer radius
2
r . The sphere is subjected to the uniform
temperature distribution
p
t and, from the moment of time 0


, to the convective-radiation
heat exchange trough the surfaces
1
rr

and
2
rr

with environments of constant
temperatures
1c

( ) ( 1) ( )( ) ( )( ) 0
j
j
tjcjjcj
rr
t
ttttttt
r









(1,2)j

, (73)

0
p
tt



. (74)
Let us construct the solution to the problem (72)–(74) for the material with simple nonlinearity
(()()const)

Fo ar

 ,
()
00
Bi
j
j
at
r


 (Biot number), and
()
3
00 0
Sk
j
j
at
rt



(Starc number). Then the problem (72)–(74) takes the dimensionless form

2
2
1
() ()







     




, (76)

Fo 0
p
TT


, (77)

Heat Conduction Problems of Thermosensitive Solids under Complex Heat Exchange

147
where
0cj cj
Ttt
. By application of the Kirchhoff transformation (9) to the nonlinear
problem (75)–(77), the following problem for




, (79)

Fo 0
0


 (80)
is obtained, where

  


4
()
4
() Bi ()(() ) Sk () ()
j
jj
c
jjj
c
j
QT T T T T T T
 

 . (81)
The heat conduction equation for the Kirchhoff’s variable

is linear, meanwhile the

j
)
1
1
(Fo) ( ) (Fo Fo )
j
m
jj jj
iii i
i
QQ QQS




  

, (82)


() () () () ()
44
( ) Bi ( )( ) Sk ( )(( ) )
jjjjj
j j cj j j cj
iiiii
QT T T T T T T
 

 , (83)

()
1, 0
S










is asymmetric unit function (H. Korn & T. Korn, 1977;
Podstrihach et al., 1984),
Fo
(
j
)
i
are the points of segmentation of the time axis
(0;Fo)
. After
substitution of the expression (82) into the boundary conditions (79), the boundary value
problem (78)–(80) becomes linear. For its solving, the Laplace integral transformation can be
used (Ditkin & Prudnikov, 1975). As a result, the Laplace transforms of the Kirhoff’s
variables are determined as
1
(1)
1






Heat Conduction – Basic Research

148

2
(2)
1
Fo
(2) (2) (2)
2
1
2
1
1
1
()
()
()
i
m
s
ii




   ;
12
() ( 1)
sh s
ss s chs
s










; s is the
parameter of Laplace transformation;
Fo
0
Fo
s
ed









11
Fo ) (Fo Fo )
() ()
ii
S





2
1
(2) (2) (2) (2) (2)
2
21 1
1
1
1
(,Fo) ( ) (,FoFo)(FoFo)
m
ii i i
i
QQQ S
 








 




;

2
()
12
2222
12 12
sin( )
2(1 )
cos( )
(1 3 )cos
j
n
j
n
njjn
n
nnn
A



 . Then on the basis of formula (9),



12
1(1 )2
p
Tk kT k


. (88)
The determined temperature is a function of coordinate

and time Fo ; it contains
12
2( )mm
approximation parameters:
1
m
values of the temperature
(1)
i
T
on the surface
1


 (due to the expressions of
(1)

in (88), the expression
of the temperature on the surface
j



are determined as

Heat Conduction Problems of Thermosensitive Solids under Complex Heat Exchange

149

12
1(1 )2 (1,2)
jj
p
Tk kTk j
 




 


. (89)
If the values
(1)
Fo
i

1
m values of
(1)
i
T and
2
m values of
(2)
i
T :

1
(1)
Fo Fo
2
(2)
Fo Fo
(1)
12
1
1
(2)
12
2
1
1(1 )2 (1, 1),
1(1 )2 (1, 1).
i
i
p




 






(90)
After solving this set of equations and substituting the values
()
(1,2)
j
i
Tj into (88), the
expression for the temperature can be obtained.
For approximation of the nonlinear expressions

()
()
j
QT

, we use the same segmentation
of the time axis
12
(mmm


Fo Fo ) contain four
values
(1)
i
T and
(2)
(2,3)
i
Ti , etc.; in the last two equations (obtained from (90) at
1
Fo Fo
m
 ) , all 2( 1)m

unknown values
(1)
i
T and
(2)
(2,)
i
Ti m are presented. After
solving the first and second equations, the values
(1)
2
T and
(2)
2
T are determined. After
substitution of these values into the third and fourth equations, the following two unknown

QQ

,
2cc
TT

, the following expression for the Kirchhoff’s
variable

1
111
1
(,Fo) ( )( ,FoFo)(FoFo)
m
ii i i
i
QQQ S
 



     

(91)

Heat Conduction – Basic Research

150
can be obtained for the solid sphere, where
2


 
1
p
TT

, and
n

are roots of the equation

t
g



. (92)
The unknown parameters
1
Fo Fo
(,Fo)
i
i
TT





are determined from the equations



, then formula (91) yields



1
1
1
Bi( )(,Fo) (,FoFo)(FoFo)
m
pc i i i i
i
TT T T S
 





  




. (94)
The unknown parameters of spline approximation
(2,)
i
Ti m are determined from the set

1Bi(1,0); 2
pp
LTkT

    . Then the solutions of second, third, and all the
following equations can be written as
2
2
011
1
1
[{ 2 Bi( )(1,Fo) (
i
ipcij
j
TLLk TT T
k








    











, (97)
the substitution of the nonlinear expression
()T

by

(Nedoseka, 1988; Podstrihach &
Kolyano, 1972) can be employed. Then the Kirchhoff’s variable can be given as

Heat Conduction Problems of Thermosensitive Solids under Complex Heat Exchange

151

2
Fo
1
sin cos
2
1sin
(sincos)
n
nn n
cn
nn n n

the time-variation of the temperature on the surface 1


of solid sphere exposed to the
condition of convective heat exchange. We assume
c
t = 300ºС (573 K) and this value is also
chosen to be the reference temperature; the initial temperature is
p
t

20ºC (293 K); the Biot
number is Bi 10
 . In the expression () (1 )
tto
tkT



 we set
to


50,2 W/(mºK) and
0,018
k  . The results of computation are shown in Figure 1. Fig. 1. Dependence of
()T

physically improper results. As it follows from the figures, at some moments of time, the
temperature on surface of sphere is greater than the temperature of heating environment.
The authors (Nedoseka, 1988; Podstrihach & Kolyano, 1972) did not give much attention to
this matter because mainly they considered the temperature fields in thermosensitive bodies
due to the internal heat sources. In this case increasing of the temperature is unbounded.
6. Conclusion
In this chapter, the formulations of non-linear heat conduction problems for the bodies with
temperature-dependent characteristics (thermosensitive bodies) are given. The efficient
analytico-numerical methods for solution of the formulated problems are developed.
Particularly, the step-by-step linearization method is proposed for solution of one-
dimensional transient problems of heat conduction, which describe the temperature fields in
thermosensitive structure members of simple nonlinearity under complex (convective,
radiation or convective-radiation) heat exchange boundary conditions. The coefficient of
heat exchange and emissivity of the surface, that is under heat exchange with environment,
are also dependent on the temperature. The method provides:
-
reduction of the heat conduction problem to the corresponding dimensionless
problem;
-
partial linearization of the obtained problem by means of the Kirchhoff’s transform;
-
complete linearization of the nonlinear condition on the Kirchhoff’s variable

, that
has been obtained from the condition of complex heat exchange due to approximation
of the nonlinear term by specially constructed spline of zero or first order;
-
construction of the solution to the linearized boundary value problem for

by means




with unknown parameter

. This parameter can be found by
satisfaction of the nonlinear condition for

with required accuracy.
The method of linearizing parameters is adopted to solution of the nonlinear steady-state
and transient heat conduction problems for contacting thermosensitive bodies of simple
geometrical shape under conditions of the ideal thermal contact at the interfaces and
complex heat exchange on the limiting surfaces. Its approbation is provided for the
n-layer
cylindrical pipe under given temperatures on its inner and outer surfaces. It these surfaces
are subjected to the convective heat exchange, then the complete linearization of the
obtained nonlinear conditions for the Kirchhoff’s variable

can be done by means of the
method of linearizing parameters.
7. Acknowledgment
This research is provided under particular support of the project within the joint program of
scientific research between the Ukrainian National Academy of Sciences and Russian
Foundation of Basic Research (2010-1011).

8. References
Carslaw, H.S. & Jaeger, J.C. (1959). Conduction of Heat in Solids, Clarendon, ISBN: 978-0-19-
853368-9, Oxford, UK
Ditkin, V.A. & Prudnikov, A.P. (1975).
Operational Calculus, Vysshaja shkola, Moscow, Rusia

pp. 357-369, ISSN 0022-0833
Kushnir, R. & Protsiuk, B. (2009). A Method of the Green’s Functions for Quasistatic
Thermoelasticity Problems in Layered Thermosensitive Bodies under Complex
Heat Exchang. In:
Operator Theory: Advances and Applications, Vol.191, V.Adamyan,

Heat Conduction – Basic Research

154
Yu.Berezansky, I.Gohberg & G.Popov, (Eds.), 143-154, Birkhauser Verlag, ISBN 978-
3-7643-9920-7, Basel, Switzerland
Lykov, A.V. (1967).
Heat Conduction Theory, Vysshaja shkola, Moscow, Rusia (in Russian)
Nedoseka, A.Ya. (1988).
Fundamentals of Design Computation for Welded Structures, Vyscha
shkola, Kyiv, Ukraine (in Russian)
Noda, N. (1986). Thermal Stresses in Materials with Temperature-Dependent Properties, In:
Thermal Stresses I, R.B. Hetnarski, (Ed.), 391-483, North-Holland, Elsevier, ISBN
0444877282 , Amsterdam, Netherland
Nowinski, J. (1962). Transient Thermoelastic Problem for an Infinite Medium with a
Spherical Cavity Exhibiting Temperature-Dependent Properties.
Journal of Applied
Mechanics,
Vol.29, pp. 399-407, ISSN: 0021-8936
Podstrihach, Ya.S. & Kolyano, Yu.M. (1972).
Nonstationary Temperature Fields and Stresses in
Thin Plates
, Naukova Dumka, Kyiv, Ukraine (in Russian)
Podstrihach, Ya.S.; Lomakin, V.A. & Kolyano, Yu.M. (1984).
Thermoelasticity of Bodies of

Sorokin, V.G.; Volosnikova A.V. & Vjatkin S.A. (1989).
Grades of Steels and Alloys,
Mashinostroenije, Moscow, Rusia (in Russian)
0
Can a Lorentz Invariant Equation Describe
Thermal Energy Propagation Problems?
Ferenc Márkus
Department of Physics, Budapest University
of Technology and Economics
Hungary
1. Introduction
In the new technologies the development towards the small scales initiates and encourages the
reformulation of those well-known transport equations, like heat and electric conduction, that
were applied for bulk materials. The reason of it is that there are several physical evidences
for the changes of the behavior of the signal propagation as the sample size is decreasing
(Anderson & Tamma, 2006; Cahill et al., 2003; Chen, 2001; Liu & Asheghi, 2004; Schwab et al.,
2000; Vázquez et al., 2009). The constructed different mathematical models clearly belong to
the phenomena of the considered systems. However, presently, there is no a well-trodden way
how to establish the required formulations in general. A great challenge is to establish and
exploit the Lagrangian and Lorentz invariant formulation of the thermal energy propagation,
since, on the one hand, the connection with other field theories including the interactions of
fields can be done on this level, on the other hand, these provide the finite physical action
and signal propagation. The results of the presented theory ensures a deeper insight into the
phenomena, thus hopefully it will contribute to the technical progress in the near future.
It is an old and toughish question how to introduce the finite speed propagation of action in
such physical processes like the thermal energy propagation (Eckart, 1940; Joseph & Preziosi,
1989; Jou et al., 2010; Márkus & Gambár, 2005; Sandoval-Villalbazo & García-Colín, 2000;
Sieniutycz, 1994; Sieniutycz & Berry, 2002). There is no doubt that the solution must exist
somehow and the suitable description should be Lorentz invariant. Moreover, this Lorentz
invariant formulation needs to involve anyway the Fourier heat conduction as the classical

Ld
3
xdt = extremum,(1)
i.e., there exists a Lagrange density function L by which the calculated action S is extremal
for the real physical processes. The Hamiltonian formulation can be also achieved for certain
differential equations involving non-selfadjoint operators like the first time derivative in the
classical Fourier heat conduction. Then such potential functions are required to introduce
by which the Lagrange functions can be expressed and the whole Hamiltionian theory can
be constructed (Gambár & Márkus, 1994; Gambár, 2005; Márkus, 2005). The long scientific
experience on this topic showed that the theories are comparable and connectable on this —
Lagrangian-Hamiltonian — level, thus in the further development of the theory it is useful
to apply this idea and scheme. In order to generate a dynamic temperature and the related
covariant Klein-Gordon type field equation, to describe the heat propagation with finite speed
— less than the speedof light — of action an abstract scalar potential field has been introduced
(Gambár & Márkus, 2007). In this case the thermal energy propagation has wave-like modes.
It is important to emphasize that, on the other hand, this scalar field can be connected to the
usual (local equilibrium) temperature and the Fourier’s heat conduction in the classical limit.
This treating is an attempt to point out that the dynamic phase transition (Ma, 1982) between
the two kinds of propagation, between a wave and a non-wave, or with another context it is
better to say — between a non-dissipative and a dissipative thermal process — has a more
general role and manifestation in the processes.
As a starting point the Lagrange functions are given for both the Lorentz invariant heat
propagation (Márkus & Gambár, 2005) and for the classical heat conduction (Fourier’s heat
conduction) (Gambár & Márkus, 1994). The first description is based on a Klein-Gordon
type equation formulated by a negative "mass term". It will be shown that this pertains
to a repulsive potential, which repulsive interaction produces a tachyon solution leading
to the so-called spinodal instability which effect is often applied in modern field theories
(Borsányi et al., 2000; 2002; 2003). Now, the Hamiltonian descriptions are written side by
side — to prepare the later comparison — showing how the Lorentz invariant solution
provides the classical solution in the limit of speed of light. The relevant Lagrangians, L

2

2
−
1
c
2
∂
2
ϕ
∂x
2
∂
2
ϕ
∂t
2
−
1
2
c
4
c
4
v
16λ
4
ϕ
2
, (2a)

is the
specific heat. Applying the calculus of variation the corresponding Euler-Lagrange equations
as equations of motion for the field ϕ can be obtained
0
=
1
c
4
∂
4
ϕ
∂t
4
+
∂
4
ϕ
∂x
4
−
2
c
2
∂
4
ϕ
∂t
2
∂x
2

temperature, which is defined from a dynamical point of view, thus it can be considered as
the dynamic temperature. Furthermore, temperature
T denotes the usual local equilibrium
temperature
T
=
1
c
2
∂
2
ϕ
∂t
2
−
∂
2
ϕ
∂x
2
+
c
2
c
2
v
4λ
2
ϕ, (4a)
T = −

c
2
c
2
v
4λ
2
T = 0, (5a)
∂
T
∂t
−
λ
c
v
∂
2
T
∂x
2
= 0. (5b)
Here, Equation (5a) — the hyperbolic one — is a Klein-Gordon type equation with a negative
"mass term"
−(c
2
c
2
v
/4λ
2

, (6a)
ω
(k)=−i
λ
c
v
k
2
= −iDk
2
.(6b)
157
Can a Lorentz Invariant Equation Describe Thermal Energy Propagation Problems?
4 Will-be-set-by-IN-TECH
Here, the diffusivity parameter D = λ/c
v
is introduced to simplify the forms. The dispersion
relation in Equation (6a) pertains to the Klein-Gordon wave equation in Equation (5a) from
which we obtain the phase velocity w
f
w
f
(k)=
ω
k
= c

1 −
c
2

−→
dω
dk




c→∞
= −i2Dk; v
T
= 2Dk  c.(8)
This limit shows clearly that the Lorentz invariant description covers both cases, and the
wave-like and the non-wave heat propagation can be discussed in the same frame.
0 1 2 3 4 5 6
0
1
2
3
4
5
6
7
x = Dk (10
8
m/s); D = 1
Im w
f
and w
f
(10

= c/2. The
value of diffusity is taken D
= 1. The phase velocity w
f
of the wave-like propagation is
always smaller than the speed of light.
It can be recognized that there is a value of the wave number k when the discriminant changes
its sign in Equation (7) at the value k
0
= c/2D. Now, the solutions can be split into two
parts. On the one hand, we can consider the case k
> k
0
, when the solution is real and
wave-like (non-dissipative), and on the other hand, we take the case k
< k
0
, when the solution
is imaginary and non-wave (dissipative). The real and the imaginary part of the phase velocity
w
f
can be written for both cases
w
f
=
ω
k
= c

1 −

field in the cosmology shown in Sec. 4) it is worthy to reformulate it for this later use. It has
been shown in the literature (Márkus & Gambár, 2005) that the quantization of the thermal
field generates quasi particles and these particles may have a mass
M
0
=
¯h
2D
, (10)
where ¯h is the Planck constant. Moreover, the Planck units are applied for the present case
(c
= 1; ¯h = 1). Then the 3D Lagrangian given by Eq. (2a) should be rewritten
L
w
=
1
2
(Δϕ)
2
+
1
2

∂
2
ϕ
∂t
2

2



∂Ψ
∂t

2
dx, (12)
where Ψ is the displacement from the equilibrium position,  is the density, A is the cross
section of the string. The mass element is dm
= Adx. The either of the potential energy
terms comes from the small deformation (elongation) of the stretching which is
V
= F

⎡
⎣

1 +

∂Ψ
∂x

2
−1
⎤
⎦
dx
∼
1
2

159
Can a Lorentz Invariant Equation Describe Thermal Energy Propagation Problems?
6 Will-be-set-by-IN-TECH
∂
2
Ψ
∂t
2
−
F
A
∂
2
Ψ
∂x
2
+
k
a
A
Ψ
= 0. (15)
can be deduced. Now, if a "repulsive" potential is imagined at the places of the springs shown
in Fig. 2(b) then a Klein-Gordon type equation with negative "mass term" (Gambár & Márkus,
2008) is obtained
∂
2
Ψ
∂t
2

The structure of this equation is exactly the same as in the case of Lorentz invariant thermal
energy propagation in Equation (5a). Since, it is clear from this mechanical example that the
negative sign of the third term in Equation (16) pertains to a repulsive interaction, thus, this is
the reson why the negative "mass term" may relate to a repulsive interaction in the relativistic
160
Heat Conduction – Basic Research
Can a Lorentz Invariant Equation Describe Thermal Energy Propagation Problems? 7
case in Equation (5a), in general. Maybe, it is complicated to prepare a device to ensure the
repulsive interaction from little springs. However, if the stretched string is placed on the
diameter of a rotating disk — shown in Fig. 2(c) that moves with the angular velocity ω
0
,then
the centrifugal force can produce the similar repulsive interaction.
The centrifugal potential of a point-like mass m moving on a circle with a radius r
−
1
2
mr
2
ω
2
0
can be generalized to the present case. This gives the potential V
rot
pertaining to the rotational
motion of the string
V
rot
= −
1

The same mathematical structure can be immediately recognized comparing this equation
with the Equations (5a) and (16). This means that these three equations must involve the
similar physical behavior: the spinodal instability and the dynamic phase transition (Gambár,
2010). All together these examples clearly prove the physical reality of the Klein-Gordon
equation with negative "mass term" in nature.
Finally, for the completeness the dispersion relation for Equation (18) can be also calculated
Ω
(k, ω
0
)=

F
A
k
2
−ω
2
0
. (19)
This formula shows again the same physical behavior clearly as it has been found in Equation
(6a). The phase velocity is
w
ph
=
Ω
k
=

F
A

,the
centrifugal force elongates the string to infinity, the string cannot have vibrating modes. The
change in the propagation modes is an angular velocity controlled dynamic phase transition
that divides the dissipative – non-dissipative transition like in Equations (7), (9a) and (9b) for
the thermal case.
161
Can a Lorentz Invariant Equation Describe Thermal Energy Propagation Problems?
8 Will-be-set-by-IN-TECH
4. Inflationary cosmology with the dynamic temperature
It is a great challenge to experience and understand how the Lorentz invariant propagating
thermal energy field ϕ can interact with other physical fields. In this way new physical
relations, considerations and explanations may be expected for the relevant phenomena.
As an advanced example, to point out the strength of the formulation, the thermal and
cosmological inflaton fields are coupled within the Lagrangian framework (Márkus et al.,
2009).
4.1 Linde’s model of the inflaton field
In the present model the cosmological model is based on the Einstein’s equation in the
Friedman-Robertson-Walker metric. Now, the action S can be expressed as
S
=


−
˜
gL
FRW
d
4
x, (23)
where the expression

−V(φ)

(24)
is the starting point in the description;
∇is the gradient operator. Then, the equation of motion
for the inflaton can be calculated
∂
2
φ
∂t
2
−
1
a
2
Δφ + 3H
∂φ
∂t
= −
δV(φ)
δφ
, (25)
where δV
(φ)/δφ means a functional derivative. The Hubble parameter H(t) is defined by
H
=
˙
a
a
. (26)

+
1
4λ
(M
2
−λσ
2
)
2
. (27)
Here, the first term on the right hand side pertains to the second term — the space derivate
term — on the left hand side in Equation (25). The second term generates the inflation
process, the third one couples the inflaton field to the introduced additional field σ and the last
one produces mass generation through the spontaneous symmetry breaking. The canonical
momentum of the inflaton field can be calculated
Π
φ
=
∂L
FRW
∂
˙
φ
=
˙
φ. (28)
Then the Hamiltonian
˜
H of the field which is the energy density can be obtained
162

H
˜
H
= 
φ
= T
00
, (30)
where T
00
is called as the time-time component of the energy-momentum tensor. Furthermore,
the Einstein’s equation can be expressed in the FRW metric as

˙
a
a

2
=
8πG
3
, (31)
where G is the gravitational constant and  is the mass density. Substituting the energy density

φ
and the Planck mass
M
pl
=


δV(φ
0
)
δφ
0
, (34)
the ’field variable’ φ
0
depends on the time parameter only. In this case the energy density 
φ
has a simplified form

φ
=

1
2

dφ
0
dt

2
+ V(φ)

, (35)
by which the equation H
2
=(1/3M
2

of cosmology requires the same mathematical frame of the description. Now, the tool is ready
163
Can a Lorentz Invariant Equation Describe Thermal Energy Propagation Problems?